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March 24, 2007

Philosophising in Brussels

Posted by David Corfield

I’m off to Brussels tomorrow. Fortunately someone pointed out to me that the clocks go forward tonight or I might have missed my train. I’m speaking at a conference called Perspectives on Mathematical Practices 2007. Here are some notes for my talk.
Posted at March 24, 2007 9:52 PM UTC

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Re: Philosophising in Brussels

Another good class of examples for complicated-looking conglomerates of data that can be realized as mere components of a single nn-categorical gadget are a plethora of various “cocycle relations” that people usually write out as long lists of symbols with, usually, confusingly many signs scattered all over the place.

Often these can be seen to be nothing but coherence conditions for structure morphisms in higher categories.

Well, I guess the encapsulation of structure is actually a recursive phenomenon:

highly complicated-looking structures may be realized as algebras for operads.

Defining an operad itself in the standard way also takes a couple of lines.

But then we follow Tom Leinster and realize that all this data going into the definition of an operad in turn may be re-assembled into the mere statement of a monad in generalized spans.

And a monad, in turn, is just a component way of saying “lax functor on the trivial 2-category”.

Re: Philosophising in Brussels

As the laws of physics only ever get modified so as to speed up time when one is giving a talk, that material on 2-class functions, inserted in case of an impossible, yet ever hoped for, slowing of time, had to be ignored.

But it would be good to collect some prize examples of this phenomenon.

Re: Philosophising in Brussels

David talked about

Mathematics organised by stories or dramatic narratives

By the way, probably you have mentioned this elsewhere, but I’d think that one reason why physics, and in particular what might be called “modern formal high energy physics” has been such a rich source of mathematical insight, at least at the level of conjectures (Witten, Kontsevich,…) is that whatever its relation to the physical world really is (which is an issue of heated – and overheated – debate, as we know #)
it does a great job of identifying lots of mathematical entities as secretly being actors in one grand story: it allows to “visualize” all kinds of sophisticated math in terms of something like physical processes.

Re: Philosophising in Brussels

Yes, I’d thoroughly agree with that.

It’s interesting, however, how easy it is to slip from there to the position that ‘purely’ mathematical intuition is quite weak so physical intuition is doing a huge amount of the work. Philosophers of physics often fail to observe that mathematics can pay back to physics by lending its intuitive stories. I was discussing this with Jacques Distler once at his blog.

Of course, we take for granted this intuitive story-like aspect of mathematics, due in large part to the expository efforts of a certain person from Riverside.

Read the post Automated Theorem ProvingWeblog: The n-Category CaféExcerpt: In Brussels, we heard from Koen Vervloesem about attempts towards better automated theorem provers. Readers of my book will know that I devoted its second chapter to automated theorem provers, to provide a relief against which to consider 'real...Tracked: April 5, 2007 12:40 PM